Clique Clustering Yields a PTAS for Max-Coloring Interval Graphs

摘要

We are given an interval graph where each interval has a weight . The goal is to color the intervals with an arbitrary number of color classes such that is minimized. This problem, called max-coloring interval graphs or weighted coloring interval graphs, contains the classical problem of coloring interval graphs as a special case for uniform weights, and it arises in many practical scenarios such as memory management. Pemmaraju, Raman, and Varadarajan showed that max-coloring interval graphs is NP-hard [21] and presented a 2-approximation algorithm. We settle the approximation complexity of this problem by giving a polynomial-time approximation scheme (PTAS), that is, we show that there is an -approximation algorithm for any . The PTAS also works for the bounded case where the sizes of the color classes are bounded by some arbitrary k = n.